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Unformatted text preview: Existence and uniqueness of solution to an ODE Jonathan L.F. King University of Florida, Gainesville FL 326112082, USA [email protected] Webpage http://www.math.ufl.edu/ ∼ squash/ 29 March, 2011 (at 11:22 ) Abstract: Exposition of the standard contraction mapping proof for a 1 storder DE. Shows how to apply this to higherorder DEs. ( The Fundamental Thm of ODEs , abbreviated FTODE : ) Entrance. Consider ♥ 1 an initialconditionDE x ( t ) = K t ; x ( t ) , where x (5) = . 1 : Here x is the unknown function . Putting sufficient conditions on K (), the kernel , will guarantee a local solution; a unique local soln. The Setting Fix a Banach space H with its zero vector; use d·e for its norm. Let B be a closed ball cen tered at , with Radius( B ) > 0. ( Possibly B will have infiniteradius and be all of H . Typically, however, B will have finite radius and H will be some Euclidean space R × N ; this will guarantee that B is compact. ) The “contraction space” Ω . Consider a com pact interval J w := [ 5 w, 5+ w ] , where posreal w is what we’ll call its “ width ” . Let Ω w be the space of continuous functions z : J w → B such that z (5) = . ♥ 1 Phrases: WLOG : ‘ Without loss of generality ’. TFAE : ‘ The following are equivalent ’. OTForm : ‘ of the form ’. FTSOC : ‘ For the sake of contradiction ’. Use iff : ‘ if and only if ’. IST : ‘ It Suffices to ’ as in ISTShow , ISTExhibit . Use w.r.t : ‘ with respect to ’ and s.t : ‘ such that ’....
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 Fall '07
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 Math, Calculus, Continuous function, Metric space, H1, Prof. JLF King, ) ﬁxedpoint., fz − fy

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